We give a criterion for unlimited growth with positive probability for a large class of multidimensional stochastic models. As a by-product, we recover the necessary and sufficient conditions for recurrence and transience for critical multitype Galton–Watson with immigration processes and also significantly improve some results on multitype size-dependent Galton–Watson processes.
"Criterion for unlimited growth of critical multidimensional stochastic models." Adv. in Appl. Probab. 48 (4) 972 - 988, December 2016.